Lesson Notes By Weeks and Term v5 - Grade 10

Lesson Video

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Performance objectives

• Welcome back to Mathematics! This week, we delve deeper into the fascinating world of functions. Last week, we introduced the basic concepts of relations, functions, domain, and range. This week, we'll focus on understanding and representing functions in different ways, specifically focusing on function notation, evaluating functions, and plotting simple functions. Understanding functions is crucial because they are mathematical models that describe relationships between quantities. They are used extensively in fields like science, engineering, economics, and even in everyday applications like calculating mobile data usage or the cost of a taxi ride.
• Why is this important for South African learners? Think about budgeting your pocket money (input: amount of money; output: items you can afford), understanding cellphone contracts (input: data usage; output: cost), or even predicting the spread of diseases (input: initial infection rate; output: number of infected people over time). Functions are the tools that allow us to analyze and understand these situations.
• By the end of this week, you will be able to:
• Objective 1: Use function notation, f(x)*, correctly to represent functions.
• Objective 2: Evaluate functions for given input values (including integers, fractions, and simple algebraic expressions).

Lesson summary

Welcome back to Mathematics! This week, we delve deeper into the fascinating world of functions. Last week, we introduced the basic concepts of relations, functions, domain, and range. This week, we'll focus on understanding and representing functions in different ways, specifically focusing on function notation, evaluating functions, and plotting simple functions. Understanding functions is crucial because they are mathematical models that describe relationships between quantities. They are used extensively in fields like science, engineering, economics, and even in everyday applications like calculating mobile data usage or the cost of a taxi ride.

Lesson notes

2.1 Function Notation Function notation is a way of writing functions that makes it easy to identify the input and output. Instead of writing y = 2x + 3, we write f(x) = 2x + 3. f(x)* is read as "f of x." f is the name of the function. x is the input variable (also called the independent variable). f(x) represents the output value (also called the dependent variable). It is equivalent to y.

Example: If f(x) = x² + 1, this means the function f takes an input x, squares it, and then adds 1. 2.2 Evaluating Functions Evaluating a function means finding the output value for a given input value. To do this, we substitute the input value for x in the function's equation.

Example 1: Let f(x) = 3x -

2. Find f(4). This means we need to substitute x = 4 into the function. f(4) = 3(4) - 2 = 12 - 2 = 10 Therefore, f(4) =

1

0. This means when the input is 4, the output is

1

0. Example 2: Let g(x) = x² - 5x +

6. Find g(-2). Substitute x = -2 into the function. g(-2) = (-2)² - 5(-2) + 6 = 4 + 10 + 6 = 20 Therefore, g(-2) =

2

0. Example 3: Let h(x) = (x + 1) / (x - 3). Find h(5). Substitute x = 5 into the function. h(5) = (5 + 1) / (5 - 3) = 6 / 2 = 3 Therefore, h(5) = 3. 2.3 Plotting Linear Functions A linear function is a function that can be written in the form f(x) = mx + c, where m is the slope and c is the y-intercept. The graph of a linear function is a straight line.

Slope (m): The slope represents the rate of change of the function. It tells us how much the output changes for every one unit change in the input. A positive slope indicates an increasing function (the line goes up from left to right), and a negative slope indicates a decreasing function (the line goes down from left to right).

Y-intercept (c): The y-intercept is the point where the line crosses the y-axis. It is the value of f(x) when x =

0. So f(0) = c.

Steps to plot a linear function: Choose two values for x. Pick easy numbers to work with! For example, x = 0 and x = 1 are often good choices. Calculate the corresponding f(x) values by substituting the x values into the function's equation. This gives you two points (x, f(x)). Plot the two points on the Cartesian plane. Draw a straight line through the two points. This line is the graph of the function.

Example: Plot the function f(x) = 2x -

1. Choose x values: Let's choose x = 0 and x =

2. Calculate f(x) values: f(0) = 2(0) - 1 = -

1. So, the point is (0, -1). f(2) = 2(2) - 1 = 4 - 1 =

3. So, the point is (2, 3). Plot the points (0, -1) and (2, 3) on a Cartesian plane. Draw a straight line through the two points. 2.4 Domain and Range of Simple Linear Functions Domain: The domain of a function is the set of all possible input values (x) for which the function is defined. For most linear functions, the domain is all real numbers because you can substitute any number for x. We write this as x ∈ ℝ (x is an element of the set of Real numbers).

Range: The range of a function is the set of all possible output values (f(x) or y) that the function can produce. For most linear functions, the range is also all real numbers. We write this as f(x) ∈ ℝ or y ∈ ℝ. 2.5 Interpreting the y-intercept The y-intercept is the point where the graph of the function crosses the y-axis. It represents the value of the function when x =

0. In real-world applications, the y-intercept often represents the initial value or starting point of a situation. For example, if f(x) = 5x + 20 represents the amount of money (in Rands) in a savings account after x weeks, the y-intercept of 20 represents the initial amount of money (R20) in the account before any weeks have passed. Guided Practice (With Solutions)

Question 1: If f(x) = -x + 5, find f(-3).

Solution: Substitute x = -3 into the function: f(-3) = -(-3) + 5 = 3 + 5 = 8 Therefore, f(-3) =

8. Commentary: This question tests the basic understanding of function evaluation. Remember to pay attention to signs, especially when dealing with negative numbers.

Question 2: Let g(x) = x² -

4. Find g(a + 1).

Solution: Substitute x = a + 1 into the function: g(a + 1) = (a + 1)² - 4 Expand the square: g(a + 1) = (a² + 2a + 1) - 4 Simplify: g(a + 1) = a² + 2a - 3 Therefore, g(a + 1) = a² + 2a -

3. Commentary: This question introduces evaluating functions with algebraic expressions as inputs. It requires careful algebraic manipulation, specifically expanding brackets.

Question 3: Plot the function h(x) = -2x + 3 on a Cartesian plane.

Solution: Choose x values: Let's choose x = 0 and x =

1. Calculate h(x) values: h(0) = -2(0) + 3 =

3. So, the point is (0, 3). h(1) = -2(1) + 3 = -2 + 3 =

1. So, the point is (1, 1). Plot the points (0, 3) and (1, 1) on a Cartesian plane. Draw a straight line through the two points. The line will slope downwards from left to right.

Commentary: This question tests the ability to plot a linear function. Choosing x=0 is always a good starting point as it immediately gives you the y-intercept.

Question 4: Determine the domain and range of the function f(x) = x/2 + 1.